246 CENTEAL FOBCES; 



parallel to S Y, and S A the semi-conjugate axis ; S A is a 

 mean proportional between S Y and P N, and therefore S Y = 



AS 2 



; also the radius of curvature of the ellipse is (like that 



4PN 8 

 of all conic sections) equal to ^ , P being the parameter. 



Therefore we have to substitute these values for S Y and the 

 radius of curvature, K, in the expression for the central force, 



SP SP P 2 



and we have ^ R = =- x SP; so 



RXSY"'" u 4.PN AS 6 ~4AS' 



P 8 



P 2 



that, neglecting the constant , the centripetal force is as 



4 A. o 



the distance directly. 



From hence it follows, conversely, that if the centripetal 

 force is as the distance, the orbit is elliptical or circular, for 

 by reversing the steps of the last demonstration we arrive at 

 an equation to the ellipse ; or, in case of the two axes being 

 equal, to the circle. It also follows that if bodies revolve in 

 circular or elliptical orbits round the same centre, the centre 

 of the figures being the centre of forces, and the force being 

 as the distance, the periodic time of all the bodies will be the 

 same, and the spaces through which they move, however dif- 

 fering in length from each other, will all be described in the 

 same time. This proposition, which sonaetimes has appeared 

 paradoxical to those who did not sufficiently reflect on the 

 subject, is quite evident from considering that the force and 

 velocity being increased in proportion to the distance, and 



